Representations for Cyclic Groups

As noted earlier, a cyclic group is Abelian, and each of its h elements is in a separate class. Therefore, it must have h one-dimensional irreducible representations. To obtain these there is a perfectly general scheme which is perhaps best explained by an example. It will be evident that the example may be generalized. Let us consider the group C5, consisting of the five commuting operations C5, C, C, C5, E we seek a set of five one- 

The remaining columns follow from the group multiplications. It will now be shown that these representations satisfy the orthonormalization condition of 4.5-1. 

Consider any two representations, say P and I where q - p - r. The left-hand side of 4.5-1 takes the form 

It is then clear that the representations are normalized, because if P = I r = 0 and 4.5-3b is simply five times e° = 1, namely, 5. 

If we now replace all n s, such as 5, 10, 15,. . . , which are equal to unity by the number 1, and reduce all other exponents in excess of 5 to their lowest values as indicated in 4.5-4, we can rewrite the table in the following form  

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